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WC from Astra Walker. In-wall cisterns fro Geberit. Lucia gooseneck sidelever mixer from Abey. Return to Habitusliving. He introduced several interesting approximations in his kinetic theory model to develop theories of both electrical and thermal conductivity of metals. First, the electron gas was assumed to be free and independent in the sense that there was no electron—electron or electron—ion interactions.

An electron moved in a straight line as per Newton's law in the absence of an external electric field. Second, when an external field is applied, the electron continues to move in a straight line between collisions with the ions the collisions with other electrons were neglected in his theory. The velocity of an electron would change immediately after a collision, which was assumed to be an instantaneous event. Fourth, immediately after each collision, an electron's velocity was not related to the velocity with which it was traveling before the collision.

The new speed of the electron depended on the temperature at the time of collision, and the direction of the new velocity was randomly directed.

Classical Gas

In a sense, the electron achieved thermal equilibrium with its surroundings. The two major achievements of Drude's model were the derivation of electrical and thermal conductivity see problems in a very simplistic way. However, it had several drawbacks. Some of these were solved by Sommerfeld, who recognized the fact the free electrons in a metal were not classical but quantum gas, for which the application of Fermi—Dirac statistics instead of Maxwell—Boltzmann statistics was appropriate.

As shown by Pathria [ 8, p. Then the distribution function. The consolidated formula. Figure 21—2. Following Pathria [ 8, p. From Eq. For a system with very large volume, the energy levels are quasi-continuous and we can replace the summation with integration over k according to Eq. Thus we obtain.


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This result is familiar for a classical ideal gas see Eq. Nils Dalarsson, These mathematical relations are used to simplify the quantitative analysis of state changes for such thermodynamic systems.

The Ventures: Classical Gas

Let us first recall the equation for the differential of the internal energy per unit mass of a thermodynamic system. In thermodynamics, it is, therefore, customary to study the changes of state of a system using the T - S and p - V diagrams, as shown in Fig. Figure 9. Change of state in T-S and p-V planes. The two pairs of variables p, v and T, s are sometimes called the canonically conjugated variables. The thermodynamic coefficients are defined as the derivatives of one of the four variables p, v, T, s with respect to any other of these four variables.

In total, there are 12 such coefficients and they can be arranged in the following rectangular array. Here, it can be shown that only 3 of these 12 coefficients are independent.

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In other words, there are nine equations that relate these 12 coefficients to each other. In order to find these nine equations, we first note that the array 9. For example, the product of the three coefficients in the first row of the array 9. Thus, we immediately obtain the first four relations between the twelve coefficients listed in 9. In order to find the remaining four relations between the thermodynamic coefficients, we use the result 9.

We have now finally established all nine relations 9. These three independent thermodynamic coefficients can be determined either experimentally phenomenological thermodynamics or using the partition function Z statistical thermodynamics. For example, we can choose the three coefficients in the upper left corner of the array 9.

We can then use the thermal-state equation 8. As a final remark, it should be noted that it is not possible to choose the three coefficients from the same row of the array 9. In other words, if we know the two coefficients from a given row, the third coefficient from the same row is uniquely determined by the equations 9. According to this model, each atom contributes z v valence electrons to a sea of electrons that are shared by the remaining ion cores. Interactions among the valence electrons as well as interactions with the ion cores are treated only on average.

Specifically, one assumes that each valence electron experiences an effective potential that is constant and set equal to zero for convenience within the metal. The potential outside the metal is assumed to be sufficiently large that the electrons are confined to the volume V of the metal.

app-staging.mavericksnacks.com/8007.php Thus, each valence electron behaves as if it were free but confined to a box of volume V. We shall see that the valence electrons constitute a very dense gas, typically , times more dense than a classical gas , so quantum effects are important. Even though the free electron model is quite naive, it works rather well for some elements, especially the alkali metals. The quantum statistics of such an electron gas are governed by the Fermi-Dirac distribution function Eq.

This free electron model of a metal was the first to explain why an electron gas in a metal contributes only a small fraction of the heat capacity that it would if it were a classical gas. We estimate the number density of an electron gas in a metal. The number n of free electrons per unit volume is. This should be compared to the number density n c of a classical ideal gas at standard temperature and pressure, where one mole occupies We see that the electron gas has a number density that is about times that of a classical gas.